Introduction to R

Macquarie University
omidreza.ghasemi@hdr.mq.edu.au

University of Western Australia
mahdi.mazidisharafabadi@research.uwa.edu.au

This document is the summary of the Introduction to R workshop.

All correspondence related to this document should be addressed to:

Omid Ghasemi (Macquarie University, Sydney, NSW, 2109, AUSTRALIA)

Email: omidreza.ghasemi@hdr.mq.edu.au

1 Research Question

The aim of the study is to test if simple arguments are more effective in belief revision than more complex arguments. To that end, we present participants with an imaginary scenario (two alien creatures on a planet) and a theory (one creature is predator and the other one is prey) and we ask them to rate the likelihood truth of the theory based on a simple fact (We adapted this method from Gregg et al.,2017; see the original study here). Then, in a between-subject manipulation, participants will be presented with either 6 simple arguments (Modus Ponens conditionals) or 6 more complex arguments (Modus Tollens conditionals), and they will be asked to rate the likelihood truth of the initial theory on 7 stages.

The first stage is the base rating stage. The next three stages include supportive arguments of the theory and the last three arguments include disproving arguments of the theory. We hypothesized that the group with simple arguments shows better persuasion (as it reflects in higher ratings for the supportive arguments) and better dissuasion (as it reflects in lower ratings for the opposing arguments).

In the last part of the study, participants will be asked to answer several cognitive capacity/style measures including CRT, AOT-E, mindware, and numeracy scales. We hypothesized that cognitive ability, cognitive style, and open-mindedness are positive predictors of persuasion and dissuasion. These associations should be more pronounced for participants in the group with complex arguments because the ability and willingness to engage in deliberative thinking may favor participants to assess the underlying logical structure of those arguments. However, for participants in the simple group, the logical structure of arguments is more evident, so participants with lower ability can still assess the logical status of those arguments.

Thus, our hypotheses for this experiment are as follows:

• Participants in the group with simple arguments have higher ratings for supportive arguments (They are more easily persuaded than those in the group with complex arguments).

• Participants in the group with simple arguments have lower ratings for opposing arguments (They are more easily dissuaded than those in the group with complex arguments).

• There are significant associations between CRT, AOT-E, Numeracy, and mindware with both persuasion and dissuasion indexes in each group and in the entire sample. The relationship between these measures should be stronger, although not significantly, for participants in the group with complex arguments.

2 Getting Ready

First, we need to design the experiment. For this experiment, we use online platforms for data collection. There are several options such as Gorilla, JSpsych, Qualtrics, psychoJS (pavlovia), etc. Since we do not need any reaction time data, we simply use Qualtrics. For an overview of different lab-based and online platforms, see here.

Next, we need to decide on the number of participants (sample size). For this study, we do not sue power analysis since we cannot access more than 120 participants. However, it is highly suggested calculate sample size using power estimation. You can find some nice tutorials on how to do that here, here, and here.

After we created the experiment and decided on the sample size, the next step is to preresigter the study. However, it would be better to do a pilot with 4 or 5 participants, clean all the data, do the desired analysis, and then pre-register the analysis and those codes. You can find the preregistration form for the current study here.

Finally, we need to restructure our project in a tidy folder with different sub-folders. Having a clean and tidy folder structure can save us! There are different formats of folder structure (for example, see here and here), but for now, we use the following structure:

3 Introduction to R

# load libraries
library(tidyverse)
library(here)
library(janitor)
library(broom)
library(afex)
library(emmeans)
library(knitr)
library(kableExtra)
library(ggsci)
library(patchwork)
library(skimr)
# install.packages("devtools")
# devtools::install_github("easystats/correlation")
library("correlation")
options(scipen=999) # turn off scientific notations
options(contrasts = c('contr.sum','contr.poly')) # set the contrast sum globally 
options(knitr.kable.NA = '')

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

R can be used as a calculator. For mathematical purposes, be careful of the order in which R executes the commands.

10 + 10

## [1] 20

4 ^ 2

## [1] 16

(250 / 500) * 100

## [1] 50

R is a bit flexible with spacing (but no spacing in the name of variables and words)

10+10

## [1] 20

10                 +           10

## [1] 20

R can sometimes tell that you’re not finished yet

10 +

How to create a variable? Variable assignment using <- and =. Note that R is case sensitive for everything

pay <- 250

month = 12

pay * month

## [1] 3000

salary <- pay * month

Few points in naming variables and vectors: use short, informative words, keep same method (e.g., not using capital words, use only _ or . ).

3.1 Function

Function is a set of statements combined together to perform a specific task. When we use a block of code repeatedly, we can convert it to a function. To write a function, first, you need to define it:

my_multiplier <- function(a,b){
  result = a * b
  return (result)
}

This code do nothing. To get a result, you need to call it:

my_multiplier (2,4)

## [1] 8

Fortunately, you do not need to write everything from scratch. R has lots of built-in functions that you can use:

round(54.6787)

## [1] 55

round(54.5787, digits = 2)

## [1] 54.58

Use ? before the function name to get some help. For example, ?round. You will see many functions in the rest of the workshop.

3.2 Basic Data Types in R:

function class() is used to show what is the type of a variable.

1. Logical: TRUE, FALSE can be abbreviated as T, F. They has to be capital, ‘true’ is not a logical data:

class(TRUE)

## [1] "logical"

class(F)

## [1] "logical"

2. Numeric: all numbers e.g. 5, 10.5, 11,37; a special type of numeric is “integer” which is numbers without decimal. Integers are always numeric, but numeric is not always integer:

class(2)

## [1] "numeric"

class(13.46)

## [1] "numeric"

3. Character: text for example, “I love R” or “4” or “4.5”:

class("ha ha ha ha")

## [1] "character"

class("56.6")

## [1] "character"

class("TRUE")

## [1] "character"

Can we change the type of data in a variable? Yes, you need to use the function as.---()

as.numeric(TRUE)

## [1] 1

as.character(4)

## [1] "4"

as.numeric("4.5")

## [1] 4.5

as.numeric("Hello")

## Warning: NAs introduced by coercion

## [1] NA

3.3 Data Structures in R

Vector: when there are more than one number or letter stored. Use the combine function c() for that.

sale <- c(1, 2, 3,4, 5, 6, 7, 8, 9, 10) # also sale <- c(1:10)

sale <- c(1:10)

sale * sale

##  [1]   1   4   9  16  25  36  49  64  81 100

Subsetting a vector:

days <- c("Saturday", "Sunday", "Monday", "Tuesday", "Wednesday", "Thursday", "Friday")

days[2]

## [1] "Sunday"

days[-2]

## [1] "Saturday"  "Monday"    "Tuesday"   "Wednesday" "Thursday"  "Friday"

days[c(2, 3, 4)]

## [1] "Sunday"  "Monday"  "Tuesday"

3.3.1 Exercise

Create a vector named my_vector with numbers from 0 to 1000 in it:

my_vector <- (0:1000)

mean(my_vector)

## [1] 500

median(my_vector)

## [1] 500

min(my_vector)

## [1] 0

range(my_vector)

## [1]    0 1000

class(my_vector)

## [1] "integer"

sum(my_vector)

## [1] 500500

sd(my_vector)

## [1] 289.1081

List: allows you to gather a variety of objects under one name (that is, the name of the list) in an ordered way. These objects can be matrices, vectors, data frames, even other list.

my_list = list(sale, 1, 3, 4:7, "HELLO", "hello", FALSE)
my_list

## [[1]]
##  [1]  1  2  3  4  5  6  7  8  9 10
## 
## [[2]]
## [1] 1
## 
## [[3]]
## [1] 3
## 
## [[4]]
## [1] 4 5 6 7
## 
## [[5]]
## [1] "HELLO"
## 
## [[6]]
## [1] "hello"
## 
## [[7]]
## [1] FALSE

Factor: Factors store the vector along with the distinct values of the elements in the vector as labels. The labels are always character irrespective of whether it is numeric or character. For example, variable gender with “male” and “female” entries:

gender <- c("male", "male", "male", " female", "female", "female")
gender <- factor(gender)

R now treats gender as a nominal (categorical) variable: 1=female, 2=male internally (alphabetically).

summary(gender)

##  female  female    male 
##       1       2       3

Question: why when we ran the above function i.e. summary(), it showed three and not two levels of the data? Hint: run ‘gender’.

gender

## [1] male    male    male     female female  female 
## Levels:  female female male

So, be careful of spaces!

3.3.2 Exercise

Create a gender factor with 30 male and 40 females (Hint: use the rep() function):

gender <- c(rep("male",30), rep("female", 40))
gender <- factor(gender)
gender

##  [1] male   male   male   male   male   male   male   male   male   male  
## [11] male   male   male   male   male   male   male   male   male   male  
## [21] male   male   male   male   male   male   male   male   male   male  
## [31] female female female female female female female female female female
## [41] female female female female female female female female female female
## [51] female female female female female female female female female female
## [61] female female female female female female female female female female
## Levels: female male

There are two types of categorical variables: nominal and ordinal. How to create ordered factors (when the variable is nominal and values can be ordered)? We should add two additional arguments to the factor() function: ordered = TRUE, and levels = c("level1", "level2"). For example, we have a vector that shows participants’ education level.

edu<-c(3,2,3,4,1,2,2,3,4)

education<-factor(edu, ordered = TRUE)
levels(education) <- c("Primary school","high school","College","Uni graduated")
education

## [1] College        high school    College        Uni graduated 
## [5] Primary school high school    high school    College       
## [9] Uni graduated 
## Levels: Primary school < high school < College < Uni graduated

3.3.3 Exercise

We have a factor with patient and control values. Here, the first level is control and the second level is patient. Change the order of levels, so patient would be the first level:

health_status <- factor(c(rep('patient',5),rep('control',5)))
health_status

##  [1] patient patient patient patient patient control control control
##  [9] control control
## Levels: control patient

health_status_reordered <- factor(health_status, levels = c('patient','control'))
health_status_reordered

##  [1] patient patient patient patient patient control control control
##  [9] control control
## Levels: patient control

Finally, can you relabel both levels to uppercase characters? (Hint: check ?factor)

health_status_relabeled <- factor(health_status, levels = c('patient','control'), labels = c('Patient','Control'))
health_status_relabeled

##  [1] Patient Patient Patient Patient Patient Control Control Control
##  [9] Control Control
## Levels: Patient Control

Matrices: All columns in a matrix must have the same mode(numeric, character, etc.) and the same length. It can be created using a vector input to the matrix function.

my_matrix = matrix(c(1,2,3,4,5,6,7,8,9), nrow = 3, ncol = 3)

my_matrix

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

Data frames: (two-dimensional objects) can hold numeric, character or logical values. Within a column all elements have the same data type, but different columns can be of different data type. Let’s create a dataframe:

id <- 1:200
group <- c(rep("Psychotherapy", 100), rep("Medication", 100))
response <- c(rnorm(100, mean = 30, sd = 5),
             rnorm(100, mean = 25, sd = 5))

my_dataframe <-data.frame(Patient = id,
                          Treatment = group,
                          Response = response)

We also could have done the below

my_dataframe <-data.frame(Patient = c(1:200),
                          Treatment = c(rep("Psychotherapy", 100), rep("Medication", 100)),
                          Response = c(rnorm(100, mean = 30, sd = 5),
                                       rnorm(100, mean = 25, sd = 5)))

In large data sets, the function head() enables you to show the first observations of a data frames. Similarly, the function tail() prints out the last observations in your data set.

head(my_dataframe) 
tail(my_dataframe)

Patient	Treatment	Response
1	Psychotherapy	33.89745
2	Psychotherapy	29.38243
3	Psychotherapy	39.72931
4	Psychotherapy	29.52331
5	Psychotherapy	31.41551
6	Psychotherapy	29.88476

	Patient	Treatment	Response
195	195	Medication	29.56865
196	196	Medication	33.14262
197	197	Medication	18.79003
198	198	Medication	19.76909
199	199	Medication	27.66853
200	200	Medication	17.03878

Similar to vectors and matrices, brackets [] are used to selects data from rows and columns in data.frames:

my_dataframe[35, 3]

## [1] 30.05458

3.3.4 Exercise

How can we get all columns, but only for the first 10 participants?

my_dataframe[1:10, ]

Patient	Treatment	Response
1	Psychotherapy	33.89745
2	Psychotherapy	29.38243
3	Psychotherapy	39.72931
4	Psychotherapy	29.52331
5	Psychotherapy	31.41551
6	Psychotherapy	29.88476
7	Psychotherapy	17.48139
8	Psychotherapy	30.31757
9	Psychotherapy	33.51684
10	Psychotherapy	36.06823

How to get only the Response column for all participants?

my_dataframe[ , 3]

##   [1] 33.897452 29.382434 39.729310 29.523313 31.415508 29.884756 17.481386
##   [8] 30.317566 33.516836 36.068230 30.792595 21.513371 33.806714 18.348887
##  [15] 33.054798 22.150448 22.770986 24.989311 30.280924 32.102438 27.821276
##  [22] 26.642487 24.778134 32.838337 36.324994 25.096328 18.623429 30.757799
##  [29] 24.465123 26.651556 25.455577 39.112993 30.755474 29.364076 30.054582
##  [36] 36.027307 28.686847 26.542273 28.128803 33.639594 24.451841 36.527995
##  [43] 33.022184 29.018652 29.005920 24.002321 31.785710 30.193651 33.266944
##  [50]  9.567978 22.529740 29.029709 32.251436 35.232829 29.077429 21.544015
##  [57] 22.646678 28.496008 30.708719 32.321395 31.583349 29.753448 33.300894
##  [64] 23.080794 20.266951 36.213717 34.179899 25.122828 28.583119 31.192843
##  [71] 32.969227 30.793463 29.183959 26.246097 31.617254 32.940636 29.067134
##  [78] 30.850151 24.365055 27.890829 23.704370 24.705611 26.795681 31.307061
##  [85] 28.631376 33.581304 24.083854 35.427188 35.250856 23.239270 31.822175
##  [92] 30.034579 24.552505 25.840735 29.761001 26.237582 22.670452 31.562706
##  [99] 25.293349 35.206560 28.695275 32.431917 21.880925 27.679252 25.653947
## [106] 22.737201 26.159944 20.220486 20.946632 26.029698  9.289973 26.064451
## [113] 24.304564 30.683930 26.556266 33.496884 25.841180 19.251002 16.588397
## [120] 27.000456 16.031283 27.138866 24.147680 27.212913 27.906480 34.292837
## [127] 35.100470 20.262522 20.990654 29.889986 22.533049 27.803764 26.561682
## [134] 31.628529 23.884178 21.184369 28.359092 29.159268 24.688478 15.449363
## [141] 33.893469 24.932828 22.202822 25.509302 19.248512 22.309783 30.230808
## [148] 21.728482 31.424331 25.274207 31.952769 25.360741 29.267328 36.850071
## [155] 32.811526 25.011702 22.615376 29.521667 26.174809 29.320354 17.688114
## [162] 21.223809 21.359619 30.377377 22.715665 26.214662 17.607744 28.204791
## [169] 25.483327 26.447378 23.768387 19.916506 26.164940 28.353001 17.307596
## [176] 19.123600 24.856908 16.758004 22.396095 30.393937 17.833034 26.610879
## [183] 25.258698 34.056139 37.833933 21.690515 12.691301 22.346300 26.041543
## [190] 21.096730 26.185576 30.638838 26.024942 22.527076 29.568648 33.142622
## [197] 18.790034 19.769089 27.668529 17.038779

Another easier way for selecting particular items is using their names that is more helpful than number of the rows in large data sets:

my_dataframe[ , "Response"]
# OR:
my_dataframe$Response

4 Data Cleaning

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

Now, suppose we tested 141 students. First, let’s read and check the uncleaned data:

# read the raw data
raw_data <- read_csv(here("raw_data","raw_argumentative_exp1.csv"))
head(raw_data)

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

end_date	status	ip_address	progress	duration_in_seconds	subject	recorded_date	response_id	location_latitude	location_longitude	distribution_channel	user_language	consent_form	age	gender	stage1_simple	stage2_simple	stage3_simple	stage4_simple	stage5_simple	stage6_simple	stage7_simple	stage1_complex	stage2_complex	stage3_complex	stage4_complex	stage5_complex	stage6_complex	stage7_complex	thinking1	thinking2	thinking3	openminded1	openminded2	openminded3	openminded4	openminded5	openminded6	openminded7	openminded8	group	numeracy_total	reasoning_total
24/9/20 22:02	IP Address	202.7.193.64	100	1517	subj1	24/9/20 22:02	R_1f298znjmVzcOjp	-33.85910	151.2002	anonymous	EN	I consent	18	Female								36	70	68	54	51	43	41	8	50	20	5	5	5	5	5	6	5	3	Complex	9	9
25/9/20 3:23	IP Address	220.245.220.94	100	1131	subj2	25/9/20 3:23	R_tL0A9P33Gi18I0N	-34.03680	150.6672	anonymous	EN	I consent	18	Male	50	55	55	90	75	50	35								8	10	39	6	6	6	5	6	6	5	6	Simple	9	10
27/9/20 22:59	IP Address	121.210.0.211	100	709	subj3	27/9/20 22:59	R_1LNyJhCKxTAAMOW	-33.85910	151.2002	anonymous	EN	I consent	19	Female	50	50	77	60	25	20	13								8	50	20	6	5	4	5	5	6	5	6	Simple	10	8
27/9/20 23:18	IP Address	58.179.100.109	100	949	subj4	27/9/20 23:18	R_3enxzUsEYgs5r1a	-12.63921	141.8741	anonymous	EN	I consent	27	Female								70	80	90	95	70	80	90	8	50	20	6	6	6	1	6	6	6	1	Complex	8	7
28/9/20 0:45	IP Address	120.154.53.68	100	1097	subj5	28/9/20 0:45	R_2Qzl2096a4KNE29	-33.85910	151.2002	anonymous	EN	I consent	19	Male	71	73	85	95	95	32	32								4	10	39	6	6	5	5	6	6	6	6	Simple	11	11
28/9/20 2:20	IP Address	1.129.107.6	100	880	subj6	28/9/20 2:20	R_esb71WOTQySjusF	-33.85910	151.2002	anonymous	EN	I consent	20	Female								89	100	44	55	100	50	55	8	50	20	6	6	6	5	6	6	6	6	Complex	10	10

Now, let’s do some cleanining using dplyr, tidyr and other tidyverse libraries. Finally, we will check the data:

cleaned_data <- raw_data %>% 
  filter(progress == 100) %>% # filter out unfinished participants
  select(-end_date, -status,-ip_address, -duration_in_seconds, -recorded_date:-user_language) %>% #remove some useless columns
  mutate(openminded_total= openminded1+openminded2+openminded3+openminded4+openminded5+openminded6+openminded7+openminded8) %>%# create a total score for our questionnaire
  mutate(thinking1= case_when(thinking1=='4'~ 1,T~0),
         thinking2= case_when(thinking2=='10'~ 1,T~0),
         thinking3= case_when(thinking3=='39'~ 1,T~0),
         thinking_total= thinking1 + thinking2 + thinking3) %>%
  select(-thinking1:-openminded8) %>%
  pivot_longer(cols = c(stage1_simple:stage7_simple,stage1_complex:stage7_complex),names_to = 'stage',values_to = 'truth_estimate') %>% # make our dataframe long
  #pivot_wider(names_from = stage, values_from= truth_estimate) # this code change our dataframe back to wide
  filter(!is.na(truth_estimate)) %>% #remove rows with truth_estimate == NA
  mutate(stage= gsub("_.*", "", stage)) %>%
  rename(consent= consent_form) %>% # rename a column
  #mutate_if(is.character, factor) %>%
  mutate(subject= factor(subject), # convert all characters to factor
         group = factor(group),
         stage = factor(stage))

progress	subject	consent	age	gender	group	numeracy_total	reasoning_total	openminded_total	thinking_total	stage	truth_estimate
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage1	36
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage2	70
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage3	68
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage4	54
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage5	51
100	subj1	I consent	18	Female	Complex	9	9	39	0	stage6	43

Ok, now the data is clean and tidy which means:

1. Each variable forms a column.
2. Each observation forms a row.
3. Each type of observational unit forms a table (Wickham, 2014).

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

Check the dataframe and all the data types:

str(cleaned_data)

## tibble [917 × 12] (S3: tbl_df/tbl/data.frame)
##  $ progress        : num [1:917] 100 100 100 100 100 100 100 100 100 100 ...
##  $ subject         : Factor w/ 131 levels "subj1","subj10",..: 1 1 1 1 1 1 1 45 45 45 ...
##  $ consent         : chr [1:917] "I consent" "I consent" "I consent" "I consent" ...
##  $ age             : num [1:917] 18 18 18 18 18 18 18 18 18 18 ...
##  $ gender          : chr [1:917] "Female" "Female" "Female" "Female" ...
##  $ group           : Factor w/ 2 levels "Complex","Simple": 1 1 1 1 1 1 1 2 2 2 ...
##  $ numeracy_total  : num [1:917] 9 9 9 9 9 9 9 9 9 9 ...
##  $ reasoning_total : num [1:917] 9 9 9 9 9 9 9 10 10 10 ...
##  $ openminded_total: num [1:917] 39 39 39 39 39 39 39 46 46 46 ...
##  $ thinking_total  : num [1:917] 0 0 0 0 0 0 0 2 2 2 ...
##  $ stage           : Factor w/ 7 levels "stage1","stage2",..: 1 2 3 4 5 6 7 1 2 3 ...
##  $ truth_estimate  : num [1:917] 36 70 68 54 51 43 41 50 55 55 ...

Finally, we save our data to the cleaned_data folder.

write_csv(cleaned_data, here("cleaned_data","argumentative_exp1.csv"))

5 Descriptive Statistics

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

Note: All the data that we use here is manipulated (fabricated) for teaching purpuses. In our study, we failed to find such beautiful and interesting results.

Now, let’s do some descriptive statistics. First, we can open a new script called analysis_exp1.r and read the cleaned data again.

data_exp1 <- read_csv(here("cleaned_data","argumentative_exp1.csv"))

How many participants in total?

data_exp1 %>% summarise(n= n_distinct(subject))

n
131

how many participants in each group?

data_exp1 %>% 
  group_by(subject) %>% 
  filter(row_number()==1) %>% 
  ungroup () %>% 
  group_by(group) %>% 
  count() 

group	n
Complex	65
Simple	66

Find the mean and sd for numeric variables using base R summary function:

data_exp1 %>% 
  group_by(subject) %>% 
  filter(row_number()==1) %>% 
  ungroup () %>%
  summary()

##     progress     subject            consent               age       
##  Min.   :100   Length:131         Length:131         Min.   :16.00  
##  1st Qu.:100   Class :character   Class :character   1st Qu.:18.00  
##  Median :100   Mode  :character   Mode  :character   Median :19.00  
##  Mean   :100                                         Mean   :21.15  
##  3rd Qu.:100                                         3rd Qu.:20.00  
##  Max.   :100                                         Max.   :63.00  
##     gender             group           numeracy_total   reasoning_total
##  Length:131         Length:131         Min.   : 0.000   Min.   : 4.00  
##  Class :character   Class :character   1st Qu.: 8.000   1st Qu.: 8.00  
##  Mode  :character   Mode  :character   Median :10.000   Median : 8.00  
##                                        Mean   : 8.779   Mean   : 8.45  
##                                        3rd Qu.:10.000   3rd Qu.: 9.00  
##                                        Max.   :11.000   Max.   :12.00  
##  openminded_total thinking_total      stage           truth_estimate  
##  Min.   :19.0     Min.   :0.0000   Length:131         Min.   :  0.00  
##  1st Qu.:33.5     1st Qu.:0.0000   Class :character   1st Qu.: 43.50  
##  Median :39.0     Median :0.0000   Mode  :character   Median : 61.00  
##  Mean   :38.2     Mean   :0.8092                      Mean   : 57.09  
##  3rd Qu.:43.0     3rd Qu.:1.0000                      3rd Qu.: 75.50  
##  Max.   :48.0     Max.   :3.0000                      Max.   :100.00

Alternatively, we can use base Rsummaryfunctionskimr` library:

data_exp1 %>% 
  group_by(subject) %>% 
  filter(row_number()==1) %>% 
  ungroup () %>% 
  dplyr::select (age, numeracy_total, reasoning_total, openminded_total, thinking_total) %>% 
  skimr::skim()

skim_type	skim_variable	n_missing	complete_rate	numeric.mean	numeric.sd	numeric.p0	numeric.p25	numeric.p50	numeric.p75	numeric.p100	numeric.hist
numeric	age	0	1	21.1526718	6.515630	16	18.0	19	20	63	▇▁▁▁▁
numeric	numeracy_total	0	1	8.7786260	2.274576	0	8.0	10	10	11	▁▁▁▂▇
numeric	reasoning_total	0	1	8.4503817	1.683466	4	8.0	8	9	12	▁▅▇▆▃
numeric	openminded_total	0	1	38.1984733	6.153698	19	33.5	39	43	48	▁▂▇▇▆
numeric	thinking_total	0	1	0.8091603	1.038598	0	0.0	0	1	3	▇▃▁▂▂

5.0.1 Exercise

For this exercise, we use a dataset of one of my own studies. In this study, we asked participants to guess the physical brightness of reasoning arguments and then we gave a cognitive ability test. (See the original study here). Open ghasemi_brightness_exp4.csv file and answer to the following questions:

1. How many participants did we test in total?
2. Find out how many male and female we tested.
3. Calculate mean and sd for age and cognitive ability (cog_ability).

ghasemi_data <- read_csv(here("cleaned_data","ghasemi_brightness_exp4.csv"))

ghasemi_data %>% summarise(n = n_distinct(participant)) # number of participants:200

n
200

ghasemi_data %>% group_by (participant) %>% filter (row_number()==1) %>% group_by (gender) %>% summarise(n= n()) %>% ungroup() # 183 female, 17 male

gender	n
Female	183
Male	17

ghasemi_data %>% dplyr::select (age, cog_ability) %>% skimr::skim() # mean and sd for age and cognitive ability

Data summary

Name	Piped data
Number of rows	38400
Number of columns	2
_______________________
Column type frequency:
numeric	2
________________________
Group variables

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	0	1	22.20	6.78	17	19	20	22	52	▇▁▁▁▁
cog_ability	0	1	39.55	9.46	11	34	40	46	61	▁▃▇▆▂

6 Data Visualization

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

First, we need to create a dataset with aggregated truth estimate scores over group and stage. We will use this dataset for line and bar graphs.

aggregated_data_exp1 <- data_exp1 %>%
  group_by(stage, group) %>%
  mutate(truth_estimate = mean(truth_estimate)) %>%
  ungroup()

barplot_exp1 <- aggregated_data_exp1 %>%
  ggplot(aes(x=stage, y= truth_estimate, fill=group)) +
  geom_bar(stat = "identity", position= "dodge")+
  # stat_summary(fun= mean, geom = "bar", position = "dodge")+ # can be used instead of geom_bar() for long dataframes
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_bw() + 
  scale_fill_jama() 

barplot_exp1

barplot_facet_exp1 <- aggregated_data_exp1 %>%
  ggplot(aes(x=group, y= truth_estimate, fill=stage)) +
  geom_bar(stat = "identity", position= "dodge")+
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_bw() + 
  theme(legend.position = "none",
        axis.text=element_text(size=11),
        axis.title = element_text(size = 12)) +
  facet_wrap(~stage)+
  scale_fill_jco() 

barplot_facet_exp1

lineplot_exp1 <- aggregated_data_exp1 %>%
  ggplot(aes(x=factor(stage), y= truth_estimate, group= group, color= group)) +
  geom_line(aes(linetype= group)) +
  geom_point(size= 5)+
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_classic() +
  theme(legend.position = "bottom",
        axis.text=element_text(size=11),
        axis.title = element_text(size = 12)) +
  scale_color_nejm() 

lineplot_exp1

violinplot_exp1 <- data_exp1 %>%
  ggplot(aes(x=factor(stage), y= truth_estimate, fill= group)) +
  geom_violin()+
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_bw() + 
  theme(legend.position = "bottom",
        axis.text=element_text(size=11),
        axis.title = element_text(size = 12)) +
  scale_fill_d3() 

violinplot_exp1

boxplot_exp1 <- data_exp1 %>%
  ggplot(aes(x=factor(stage), y= truth_estimate, fill= group)) +
  geom_boxplot()+
  #geom_point(position = position_dodge(width=0.75), alpha= .5)+
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_bw() + 
  theme(legend.position = "bottom",
        axis.text=element_text(size=11),
        axis.title = element_text(size = 12)) +
  scale_fill_simpsons() 

boxplot_exp1

boxplot_facet_exp1 <- data_exp1 %>%
  ggplot(aes(x=factor(stage), y= truth_estimate, fill= group)) +
  geom_boxplot()+
  labs (x= '', y= "Truth Likelihhod Estimate") + 
  theme_bw() + 
  theme(legend.position = "bottom",
        axis.text=element_text(size=11),
        axis.title = element_text(size = 12),
        axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1)) +
  facet_wrap(~group)+
  scale_color_simpsons() 

boxplot_facet_exp1

How to combine multiple plots? We can use the patchwork package. A nice tutorial on using this package can be found here

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

combined_plot_exp1 <- (barplot_facet_exp1+lineplot_exp1) / (violinplot_exp1+boxplot_exp1)
combined_plot_exp1

How to save a plot?

ggsave(combined_plot_exp1, filename = here("outputs","combined_plot_exp1.png"), dpi=300)

7 Data Analysis

7.1 t-test

Is there a difference between groups at the first stage? Ideally, we want participants’ ratings at the first stage be similar for both groups because we have not done any manipulations. Previous graphs showed us that ratings of simple and complex group at this stage are pretty close. Let’s test that using an independent t-test (because we have 2 independent groups):

# Is there a difference between groups at the first stage?
data_exp1 %>% 
  group_by(group) %>% 
  filter(stage=='stage1') %>% 
  ungroup () %>%
  t.test(truth_estimate~group, data = ., paired=FALSE)

## 
##  Welch Two Sample t-test
## 
## data:  truth_estimate by group
## t = -0.75145, df = 104.95, p-value = 0.4541
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.883716   5.351781
## sample estimates:
## mean in group Complex  mean in group Simple 
##              55.44615              58.71212

Now, we wonder if opposing arguments were effective at all, regardless of participants’ group. So, we would like to test if ratings at the final stage are lower than ratings at the stage 4? Since a pair of score at stage 4 and stage 7 is coming from a same person, we use paired t-test.

# Is there a difference between ratings of stage4 and stage7?
data_exp1 %>% 
  filter(stage=='stage4' | stage=='stage7') %>% 
  ungroup () %>%
  t.test(truth_estimate~stage, data = ., paired=TRUE)

## 
##  Paired t-test
## 
## data:  truth_estimate by stage
## t = 12.788, df = 130, p-value < 0.00000000000000022
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  32.64368 44.59296
## sample estimates:
## mean of the differences 
##                38.61832

7.1.1 Exercise

John et al. (2019) investigated the consequences of backing down (changing one’s mind in lights of evidence)and how other people view someone who change their mind. In their second experiments, they presented participants either with a person who changes their mind or a person who refuses to back down. Then, they asked participants to rate how intelligent and confident the person is (See the original study here). They reported that:

“Relative to the entrepreneur who did not back down, participants judged the entrepreneur who backed down as more intelligent (M_backed_down=5.13 out of 7, SD=1.09; M_did_not_back_down=3.97, SD=1.54; t(271.12)=−7.59, p < .001) but less confident (M_backed_down=4.50 out of 7, SD=1.36; M_did_not_back_down=5.65, SD=1.10; t(291.01)=8.08, p < .001).”.

Open the john_backdown_exp2.csv file and try to reproduce their results. Run two separate independent t-test, one with intelligent as the dependent variable and one with confident as the dependent variable. For both t-test, use back_down as the between-subject independent variable.

john_data <- read_csv(here("cleaned_data","john_backdown_exp2.csv"))


t.test(intelligent~back_down, data = john_data, paired=FALSE)

## 
##  Welch Two Sample t-test
## 
## data:  intelligent by back_down
## t = 7.5853, df = 271.12, p-value = 0.0000000000005319
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.8577107 1.4590076
## sample estimates:
##       mean in group backed_down mean in group did_not_back_down 
##                        5.129412                        3.971053

t.test(confident~back_down, data = john_data, paired=FALSE)

## 
##  Welch Two Sample t-test
## 
## data:  confident by back_down
## t = -8.0763, df = 291.01, p-value = 0.00000000000001787
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.4257768 -0.8670294
## sample estimates:
##       mean in group backed_down mean in group did_not_back_down 
##                        4.503268                        5.649671

7.2 Analysis of Variance (ANOVA)

Now, let’s answer our main question: Do participants in the simple group show higher ratings for supportive arguments (stage 2 to 4) and lower ratings for opposing arguments (stage 5 to 7), compared to participants in the complex group? If this is the case. we expect an interaction in the traditional Analysis of Variance (AONVA) test.

aov_m1 <- aov_car (truth_estimate ~ group*stage +
                     Error(subject/stage), data = data_exp1)

Effect	df	MSE	F	ges	p.value
group	1, 129	949.04	0.01	<.0001	.94
stage	4.45, 574.05	515.69	59.48 ***	.25	<.0001
group:stage	4.45, 574.05	515.69	13.34 ***	.07	<.0001

As you can see, we found a significant main effect of stage and a significant group by stage interaction. We can use the emmeans package to do post-hoc tests.

# main effect of stage
emmeans(aov_m1, 'stage')

##  stage  emmean   SE  df lower.CL upper.CL
##  stage1   57.1 1.88 763     53.4     60.8
##  stage2   66.8 1.88 763     63.1     70.5
##  stage3   74.6 1.88 763     70.9     78.3
##  stage4   79.6 1.88 763     75.9     83.3
##  stage5   62.4 1.88 763     58.7     66.1
##  stage6   52.5 1.88 763     48.9     56.2
##  stage7   41.1 1.88 763     37.4     44.8
## 
## Results are averaged over the levels of: group 
## Warning: EMMs are biased unless design is perfectly balanced 
## Confidence level used: 0.95

pairs(emmeans(aov_m1, 'stage'), adjust= 'holm')

##  contrast        estimate   SE  df t.ratio p.value
##  stage1 - stage2    -9.74 2.42 774 -4.031  0.0004 
##  stage1 - stage3   -17.53 2.42 774 -7.256  <.0001 
##  stage1 - stage4   -22.50 2.42 774 -9.311  <.0001 
##  stage1 - stage5    -5.29 2.42 774 -2.187  0.1160 
##  stage1 - stage6     4.53 2.42 774  1.876  0.1220 
##  stage1 - stage7    15.98 2.42 774  6.613  <.0001 
##  stage2 - stage3    -7.79 2.42 774 -3.225  0.0066 
##  stage2 - stage4   -12.76 2.42 774 -5.280  <.0001 
##  stage2 - stage5     4.46 2.42 774  1.844  0.1220 
##  stage2 - stage6    14.28 2.42 774  5.908  <.0001 
##  stage2 - stage7    25.72 2.42 774 10.644  <.0001 
##  stage3 - stage4    -4.97 2.42 774 -2.055  0.1206 
##  stage3 - stage5    12.25 2.42 774  5.069  <.0001 
##  stage3 - stage6    22.07 2.42 774  9.132  <.0001 
##  stage3 - stage7    33.51 2.42 774 13.869  <.0001 
##  stage4 - stage5    17.22 2.42 774  7.124  <.0001 
##  stage4 - stage6    27.04 2.42 774 11.188  <.0001 
##  stage4 - stage7    38.48 2.42 774 15.924  <.0001 
##  stage5 - stage6     9.82 2.42 774  4.064  0.0004 
##  stage5 - stage7    21.27 2.42 774  8.800  <.0001 
##  stage6 - stage7    11.45 2.42 774  4.736  <.0001 
## 
## Results are averaged over the levels of: group 
## P value adjustment: holm method for 21 tests

# group by stage interaction
emmeans(aov_m1, "group", by= "stage")

## stage = stage1:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   55.4 2.67 766     50.2     60.7
##  Simple    58.7 2.65 761     53.5     63.9
## 
## stage = stage2:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   63.3 2.67 766     58.1     68.6
##  Simple    70.3 2.65 761     65.1     75.5
## 
## stage = stage3:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   70.0 2.67 766     64.7     75.2
##  Simple    79.3 2.65 761     74.1     84.5
## 
## stage = stage4:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   71.6 2.67 766     66.3     76.8
##  Simple    87.6 2.65 761     82.4     92.8
## 
## stage = stage5:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   64.2 2.67 766     58.9     69.4
##  Simple    60.5 2.65 761     55.3     65.8
## 
## stage = stage6:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   57.9 2.67 766     52.7     63.2
##  Simple    47.2 2.65 761     41.9     52.4
## 
## stage = stage7:
##  group   emmean   SE  df lower.CL upper.CL
##  Complex   51.1 2.67 766     45.9     56.4
##  Simple    31.1 2.65 761     25.9     36.3
## 
## Warning: EMMs are biased unless design is perfectly balanced 
## Confidence level used: 0.95

update(pairs(emmeans(aov_m1, "group", by= "stage")), by = NULL, adjust = "holm") 

##  contrast         stage  estimate   SE  df t.ratio p.value
##  Complex - Simple stage1    -3.27 3.76 763 -0.868  0.6673 
##  Complex - Simple stage2    -6.96 3.76 763 -1.851  0.1935 
##  Complex - Simple stage3    -9.29 3.76 763 -2.469  0.0550 
##  Complex - Simple stage4   -16.02 3.76 763 -4.259  0.0001 
##  Complex - Simple stage5     3.64 3.76 763  0.967  0.6673 
##  Complex - Simple stage6    10.79 3.76 763  2.868  0.0213 
##  Complex - Simple stage7    20.08 3.76 763  5.337  <.0001 
## 
## P value adjustment: holm method for 7 tests

You can use the afex_plot function from afex to create beautiful plots. Those plots interacts nicely with ggplot:

afex_plot(aov_m1, x = "stage", trace = "group", error='between',
          line_arg = list(size=1),
          point_arg = list(size=3.5),
          data_arg = list(size= 1, color= 'grey', width=.4),
          data_geom = geom_boxplot,
          mapping = c("linetype", "shape", "fill"),
          legend_title = "Group") +
  labs(y = "Truth Likelihhod Estimate", x = "") +
  theme_bw()+ # remove the grey background and grid
  theme(axis.text=element_text(size=13),
        axis.title = element_text(size = 13),
        legend.text=element_text(size=13),
        legend.title=element_text(size=13),
        legend.position='bottom',
        legend.key.size = unit(1, "cm"),
        legend.background = element_rect(colour = 'black', fill = 'white', linetype='solid'))+
  scale_color_simpsons() +
  scale_fill_simpsons()

If you are interested in this topic, check out this nice tutorial about using afex to run ANOVA, and also this interesting tutorial on the emmeans package.

7.2.1 Exercise

Rotello et al. (2018) investigated the association between the race (White vs. Black faces) and the gun-tool judgments. In their first experiments, they presented participants with 16 White male faces and 16 Black male faces, and following that 8 images of guns and 8 images of tools. They asked participants to judge if the object is a tool or a gun by pressing keyboard buttons. Then, they ran an ANOVA to see if participants’ gun responses are higher for any of the races. So, they included prime race (Black, White) and target identity (gun, tool) as independent variables and participants’ gun responses as dependent variable into their linear model (See the original study here). They found that:

“Participants made more gun responses to guns than to tools, F(1,45) = 53243, p < 0.0001, η2g = 0.998. However, the race of the prime face did not matter, F(1,45) = 0.287, p > 0.59, η2g = 0.001, nor was there an interaction of prime race with target object, F(1,45) = 0.022, p > 0.88, η2g = 0.000)”.

Open the rotello_shooter_exp1.csv file and try to reproduce their results. Run an ANOVA (type III) with resp as the dependent variable and target, prime, and their interaction as independent variables.

# load the general data file
rotello_data <- read_csv(here("cleaned_data","rotello_shooter_exp1.csv"))

# ANOVA
rotello_aov <- aov_car (resp ~ target*prime +
           Error(subject/target*prime), data = rotello_data)

Effect	df	MSE	F	ges	p.value
target	1, 45	0.00	53242.99 ***	>.99	<.0001
prime	1, 45	0.00	0.29	.001	.59
target:prime	1, 45	0.00	0.02	<.0001	.88

7.3 Correlation

Now, let’s answer to another question of this study: does persuasion and dissuasion is related to open-mindedness, cognitive ability, reasoning abilities, and cognitive style? To answer this question, we need to create two indexes (scores) one for persuasion and one for dissuasion. Then we can do a correlation test:

cor_data_exp1 <- data_exp1 %>% 
  pivot_wider(names_from = stage, values_from = truth_estimate) %>%
  group_by(subject) %>%
  mutate(persuasion_index= stage2+ stage3+ stage4 - stage1,
         dissuasion_index= (101-stage5) + (101-stage6) + (101-stage7) - (101-stage4)) %>%
  ungroup()%>%
  dplyr::select(persuasion_index,dissuasion_index,openminded_total,numeracy_total,thinking_total,reasoning_total)

#---------- Base R:
cor(cor_data_exp1, method = "pearson",  use = "complete.obs")

#---------- Psych library:
cor_data_exp1 %>% 
  psych::pairs.panels(method = "pearson", hist.col = "#00AFBB", density = T, ellipses = F, stars = T)

#---------- Correlation library:
correlation::correlation(cor_data_exp1) %>% summary()

#---------- apaTables library:
cor_data_exp1 %>% 
  apaTables::apa.cor.table(filename="./outputs/CorMatrix.doc", show.conf.interval=T)

	persuasion_index	dissuasion_index	openminded_total	numeracy_total	thinking_total	reasoning_total
persuasion_index	1.00	0.26	0.25	0.16	0.16	0.11
dissuasion_index	0.26	1.00	-0.03	-0.03	-0.09	0.15
openminded_total	0.25	-0.03	1.00	0.40	0.26	0.11
numeracy_total	0.16	-0.03	0.40	1.00	0.44	0.15
thinking_total	0.16	-0.09	0.26	0.44	1.00	0.29
reasoning_total	0.11	0.15	0.11	0.15	0.29	1.00

Parameter	reasoning_total	thinking_total	numeracy_total	openminded_total	dissuasion_index
persuasion_index	0.11	0.16	0.16	0.25	0.26
dissuasion_index	0.15	-0.09	-0.03	-0.03
openminded_total	0.11	0.26	0.40
numeracy_total	0.15	0.44
thinking_total	0.29

7.3.1 Exercise

Pennycook et al. (2020) investigated the relationship between actively open-minded thinking style about evidence (AOT-E) and different political, scientific, and religious beliefs (see the original paper here). In their first experiment, they calculated the correlation of AOTE and scientific beliefs items (global warming, evolution, etc.) and they found the following results:

adapted from Pennycook et al. (2020)

Open the pennycook_aote_exp1.csv file and try to reproduce their results by creating the same correlation matrix.

pennycook_data <- read_csv(here("cleaned_data","pennycook_aote_exp1.csv")) 


#---------- Base R:
cor(pennycook_data, method = "pearson",  use = "complete.obs")

#---------- Psych library:
pennycook_data %>% 
  psych::pairs.panels(method = "pearson", hist.col = "#00AFBB", density = T, ellipses = F, stars = T)

#---------- Correlation library:
correlation::correlation(pennycook_data) %>% summary()

#---------- apaTables library:
pennycook_data %>% 
  apaTables::apa.cor.table(filename="./outputs/CorMatrix.doc", show.conf.interval=T)

Parameter	trust_scien	gm_health	tech_problems	modern_medicine	old_earth	vaccines	stem_cell	big_bang	evolution	global_warming
aote	0.35	0.36	0.44	0.33	0.40	0.47	0.45	0.51	0.51	0.37
global_warming	0.42	0.06	0.14	0.18	0.33	0.26	0.31	0.33	0.38
evolution	0.48	0.33	0.28	0.36	0.47	0.39	0.54	0.78
big_bang	0.49	0.37	0.28	0.36	0.45	0.37	0.54
stem_cell	0.47	0.34	0.36	0.47	0.40	0.40
vaccines	0.43	0.52	0.49	0.53	0.38
old_earth	0.29	0.24	0.21	0.33
modern_medicine	0.43	0.42	0.47
tech_problems	0.33	0.39
gm_health	0.31

7.4 Linear Regression

In the previous section, we found that open-mindedness (AOT-E) is correlated with persuasion. Now, one may ask if open-mindedness can predict persuasion after controlling for reasoning and controlling abilities? To answer that, we can run a multiple regression analysis:

exp1_reg=lm(persuasion_index ~ openminded_total+ numeracy_total+ thinking_total+ reasoning_total,
                  data=cor_data_exp1)

term	estimate	std.error	statistic	p.value
(Intercept)	78.57	33.08	2.38	0.02
openminded_total	1.62	0.72	2.23	0.03
numeracy_total	0.72	2.11	0.34	0.73
thinking_total	3.09	4.51	0.68	0.49
reasoning_total	1.77	2.52	0.70	0.48

7.4.1 Exercise

Trémolière and Djeriouat (2020) examined the role of cognitive reflection and belief in science in climate change skepticism. In their first study, they revealed that cognitive reflection and belief in science negetively predicted climate change skepticism even after controlling for demographic and cognitive ability variables (see the original paper here).

adapted from Trémolière and Djeriouat (2020)

Open the tremoliere_data_exp1.csv file and try to reproduce their results by running a multiple linear regression. Enter age, gender, education, belief in science, literacy, numeracy (Numtotal), and cognitive reflection as predictors and enter climate change skepticism (climato) as the outcome variable.

Tremoliere_data <- read_csv(here("cleaned_data","tremoliere_data_exp1.csv"))

Tremoliere_reg=lm(Climato ~ Age+ Gender+ Education+ BeliefInSciencetotal+ Literacy+ Numtotal+ CognitiveReflection,
                    data=Tremoliere_data)

term	estimate	std.error	statistic	p.value
(Intercept)	57.57	5.19	11.09	0.00
Age	0.01	0.05	0.24	0.81
Gender	-5.68	1.34	-4.23	0.00
Education	0.54	0.38	1.43	0.15
BeliefInSciencetotal	-0.20	0.06	-3.62	0.00
Literacy	-0.49	0.51	-0.96	0.34
Numtotal	-1.52	0.83	-1.82	0.07
CognitiveReflection	-18.58	4.26	-4.37	0.00

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.19	0.17	12.65	11.91	0	7	-1467.77	2953.54	2988.81	58235.89	364	372

8 Rmarkdown

To be completed…

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

Artwork by Allison Horst: https://github.com/allisonhorst/stats-illustrations

9 References

• Ghasemi, O., Handley, S., & Howarth, S. (2020). The Bright Homunculus in our Head: Individual Differences in Intuitive Sensitivity to Logical Validity.

• John, L. K., Jeong, M., Gino, F., & Huang, L. (2019). The self-presentational consequences of upholding one’s stance in spite of the evidence. Organizational Behavior and Human Decision Processes, 154, 1-14.

• Pennycook, G., Cheyne, J. A., Koehler, D. J., & Fugelsang, J. A. (2020). On the belief that beliefs should change according to evidence: Implications for conspiratorial, moral, paranormal, political, religious, and science beliefs. Judgment and Decision Making, 15(4), 476.

• Rotello, C. M., Kelly, L. J., Heit, E., Vazire, S., & Vul, E. (2018). The Shape of ROC Curves in Shooter Tasks: Implications for Best Practices in Analysis. Collabra: Psychology, 4(1).

• Trémolière, B., & Djeriouat, H. (2020). Don’t you see that its cold! Exploring the roles of cognitive reflection, climate science literacy, illusion of knowledge, and political orientation in climate change skepticism.

• Wickham, H. (2014). Tidy data. Journal of Statistical Software, 59(10), 1-23.